UMD Theses and Dissertations

Permanent URI for this collectionhttp://hdl.handle.net/1903/3

New submissions to the thesis/dissertation collections are added automatically as they are received from the Graduate School. Currently, the Graduate School deposits all theses and dissertations from a given semester after the official graduation date. This means that there may be up to a 4 month delay in the appearance of a given thesis/dissertation in DRUM.

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    Parabolic Higgs bundles and the Deligne-Simpson Problem for loxodromic conjugacy classes in PU(n,1)
    (2017) Maschal Jr, Robert Allan; Wentworth, Richard; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    In this thesis we study the Deligne-Simpson problem of finding matrices $A_j\in C_j$ such that $A_1A_2\ldots A_k = I$ for $k\geq 3$ fixed loxodromic conjugacy classes $C_1,\ldots,C_k$ in $PU(n,1)$. Solutions to this problem are equivalent to representations of the $k$ punctured sphere into $PU(n,1)$, where the monodromy around the punctures are in the $C_j$. By Simpson's correspondence \cite{s1}, irreducible such representations correspond to stable parabolic $U(n,1)$-Higgs bundles of parabolic degree 0. A parabolic $U(n,1)$-Higgs bundle can be decomposed into a parabolic $U(1,1)$-Higgs bundle and a $U(n-1)$ bundle by quotienting out by the rank $n-1$ kernel of the Higgs field. In the case that the $U(1,1)$-Higgs bundle is of loxodromic type, this construction can be reversed, with the added consequence that the stability conditions of the resulting $U(n,1)$-Higgs bundle are determined only by the kernel of $\Phi$, the number of marked points, and the degree of the $U(1,1)$-Higgs bundle. With this result, we prove our main theorem, which says that when the log eigenvalues of lifts $\widetilde{C}_j$ of the $C_j$ to $U(n,1)$ satisfy the inequalities in \cite{biswas} for the existence of a stable parabolic bundle, then there is a stable parabolic $U(n,1)$-Higgs bundle whose monodromies around the marked points are in $\widetilde{C}_j$. This new approach using Higgs bundle techniques generalizes the result of Falbel and Wentworth in \cite{fw1} for fixed loxodromic conjugacy classes in $PU(2,1)$. This new result gives sufficient, but not necessary, conditions for the existence of an irreducible solution to the Deligne-Simpson problem for fixed loxodromic conjugacy classes in $PU(n,1)$. The stability assumption cannot be dropped from our proof since no universal characterization of unstable bundles exists. In the last chapter, we explore what happens when we change the weights of the stable kernel in the special case of three fixed loxodromic conjugacy classes in $PU(3,1)$. Using the techniques from \cite{fw2}, \cite{fw1}, and \cite{paupert}, we can show that our construction implies the existence of many other solutions to the problem.
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    Generalizations of Schottky groups
    (2017) Burelle, Jean-Philippe; Goldman, William M; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    Schottky groups are classical examples of free groups acting properly discontinuously on the complex projective line. We discuss two different applications of similar constructions. The first gives examples of 3-dimensional Lorentzian Kleinian groups which act properly discontinuously on an open dense subset of the Einstein universe. The second gives a large class of examples of free subgroups of automorphisms groups of partially cyclically ordered spaces. We show that for a certain cyclic order on the Shilov boundary of a Hermitian symmetric space, this construction corresponds exactly to representations of fundamental groups of surfaces with boundary which have maximal Toledo invariant.